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Approximate analytical closed energy formulas for semirelativistic Hamiltonians of theform σVp~2m~2-1-V(r) are obtained within the framework of auxiliary field method.This method, which is equivalent to the envelope theory, has b...
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Approximate analytical closed energy formulas for semirelativistic Hamiltonians of theform σVp~2m~2-1-V(r) are obtained within the framework of auxiliary field method.This method, which is equivalent to the envelope theory, has been recently proposed asa powerful tool to get approximate analytical solutions of the Schrodinger equation.Various shapes for the potential V(r) are investigated: power-law, funnel, square root,and Yukawa. A comparison with the exact results is discussed in detail.
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The eigenenergies ε~((N))(m; {n_i , l_i }) of a system of N identical particles with a mass m are functions of the various radial quantum numbers n_i and orbital quantum numbers l_i . Approximations E~((N))(m; Q) of these eigenen...
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The eigenenergies ε~((N))(m; {n_i , l_i }) of a system of N identical particles with a mass m are functions of the various radial quantum numbers n_i and orbital quantum numbers l_i . Approximations E~((N))(m; Q) of these eigenenergies, depending on a principal quantum number Q({n_i , l_i }), can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities E~((N))(m; Q) and E~((p))(m'; Q') for various forms of the potentials (independent ofm and N) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies ε~((N))(m; {n_i , l_i }).
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The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies of the Schr_dinger equation. This method has already been successfully applied to the case of central potentials of power-law an...
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The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies of the Schr_dinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schr_dinger equation with exponential potentials of the form _arλ exp(-βr) can also be analytically solved by using the auxiliary field method. Closed formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn to the Yukawa potential and the pure exponential potential.
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The Hellmann-Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is sho...
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The Hellmann-Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.
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In 1849, Hippolyte Fizeau determined the speed of light in a famous experiment. The idea was to measure the time taken for a pulse of light to travel between an intense light source and a mirror about 8 km away. A rotating cogwhee...
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In 1849, Hippolyte Fizeau determined the speed of light in a famous experiment. The idea was to measure the time taken for a pulse of light to travel between an intense light source and a mirror about 8 km away. A rotating cogwheel with 720 notches, that could be rotated at a variable speed, was used to chop the light beam and determine the flight time. In 2017, physicists and technicians of the University of Mons in Belgium reproduced the experiment with modern devices to allow members of the public to measure the speed of light themselves. The light source used was a low power laser, and the cogwheel was replaced by an electrically driven chopper, but the general spirit of Fizeau's experiment was preserved. The exhibition was organised in the belfry of Mons, a baroque-style building classified as a UNESCO World Heritage site. The solutions found for the main problems encountered are presented here to help colleagues intending to reproduce the experiment.
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The auxiliary field method has been recently proposed as an efficient technique to compute analytical approximate solutions of eigenequations in quantum mechanics. We show that the auxiliary field method is completely equivalent t...
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The auxiliary field method has been recently proposed as an efficient technique to compute analytical approximate solutions of eigenequations in quantum mechanics. We show that the auxiliary field method is completely equivalent to the envelope theory, which is another well-known procedure to analytically solve eigenequations, although relying on different principles a priori. This equivalence leads to a deeper understanding of both frameworks.
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An interstellar Bussard ramjet is a spaceship using the protons of the interstellar medium in a fusion engine to produce thrust. In a recent paper, it was shown that the relativistic equation of motion of an ideal ramjet (100% eff...
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An interstellar Bussard ramjet is a spaceship using the protons of the interstellar medium in a fusion engine to produce thrust. In a recent paper, it was shown that the relativistic equation of motion of an ideal ramjet (100% efficient hydrogen fusion engine) is analytical. When a radiation loss appears, the speed of the ramjet is limited. But, the parametric equations, in terms of the ramjet's speed, for the position of the ramjet in the inertial frame of the interstellar medium, the time in this frame, and the proper time indicated by the clocks on board the spaceship, can still be obtained in an analytical form. The nonrelativistic motion and the motion near the limit speed are studied.
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The resolution of the Schr?dinger equation for the translation-invariant N-body harmonic oscillator Hamiltonian in D dimensions with one-body and two-body interactions is performed by diagonalizing a matrix J of order N - 1. It ha...
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The resolution of the Schr?dinger equation for the translation-invariant N-body harmonic oscillator Hamiltonian in D dimensions with one-body and two-body interactions is performed by diagonalizing a matrix J of order N - 1. It has been previously established that the diagonalization can be analytically performed in specific situations, such as for N ≤5 or for N identical particles. We show that the matrix J is diagonal, and thus the problem can be analytically solved, for any number of arbitrary masses provided some specific relations exist between the coupling constants and the masses. We present analytical expressions for the energies under those constraints.
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A method based on the envelope theory is presented to compute approximate solutions forN-body Hamiltonians with identical particles inDdimensions (D?2). In some favorable cases, the approximate eigenvalues can be analytically dete...
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A method based on the envelope theory is presented to compute approximate solutions forN-body Hamiltonians with identical particles inDdimensions (D?2). In some favorable cases, the approximate eigenvalues can be analytically determined and can be lower or upper bounds. The accuracy of the method is tested with several examples, and an application to aN-body system with a minimal length is studied. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues.
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A method is presented to compute approximate solutions for eigenequations in quantum mechanics with an arbitrary kinetic part. In some cases, the approximate eigenvalues can be analytically determined and they can be lower or uppe...
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A method is presented to compute approximate solutions for eigenequations in quantum mechanics with an arbitrary kinetic part. In some cases, the approximate eigenvalues can be analytically determined and they can be lower or upper bounds. A semiclassical interpretation of the generic formula obtained for the eigenvalues supports a new definition of the effective particle mass used in solid state physics. An analytical toy model with a Gaussian dependence in the momentum is studied in order to check the validity of the method.
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